# Grade 10 Quadratics

### By: Ravi Seeder

## Introducing Quadratics

*Quadratics can be used to graph the path or flight of a certain object which are represented by parabola's on a graph and the name comes from 'quad' which means square. So if an equation has 'x2' you know if it is an quadratic equation.*

## Table Of Contents

**1. Quadratics Relations In Vertex Form: y=a(x-h)2+k**

-Axis Of Symmetry (x)

-Optimal Value (y)

-Transformation (Vertical Stretch/Compression)

-X-intercepts

-Step Pattern

-Word Problems

**2. Quadratic Relations In Factored Form: y=a(x-r)(x-s)**

-X-intercepts and sub/isolating

-Axis of Symmetry

-Optimal Value

-What is Factoring

-Common Factoring, Factoring Perfect Squares, Factoring Different Squares, Factoring Simple and Complex Trinomials

-Word Problems

**3.Quadratic Relations in Standard form: y= ax2 + bx + c**

-Quadratic Formula

-Axis Of Symmetry

-Optimal Value

-Completing The Square

-Factoring back Into Factored Form

-Word Problems

**4. Reflection**

- A memory of mine in quadratics.

**5. Connection**

-How these different methods/formulas become one as quadratics

-How all relates to each other.

## 1. Identifying Quadratic Relations in Vertex Form: y=a(x-h)2 +k

**Example: y=2(x-2)2 +1**

**h**= Your x-intercept. (In brackets is going to be opposite form so -2 is actually +2)

**k**= Your y-intercept.

*Your 'h' can also represent your axis of symmetry. *

*Your 'y' is also represented to be your optimal value.*

**a**= Your vertical stretch/compression. NOTE: Is your 'a' is less than 1 its is going to be compression and if it is more than one its going to be a stretch. Also Note that if your 'a' is a negative number, your parabola is going to be reflecting..

**Therefore**:

*Vertex: (2,1)*

*Max/Min: Min because your parabola is not reflecting due to 'a'*

*Axis of symmetry: x=2*

*Quadratic equation in vertex form: *

*Also Note: When you are isolating for 'x' y=0 and when your *

## -The Step Pattern.

The original step patter is 'Over 1 up 1 and Over 2 up 4'

But in quadratic relations if there is an 'a' value the step pattern changes.

The 'a' value gets multiplied with each 'up' value.

**Example: y=2(x-4)2 +1**

Due to the 'a' value in this relation being 2 you must multiply each 'up' with 2.

So the new step pattern becomes: **Over 1 up** **2** **and over 2 up** **8**

## First and Second Differences

## Word Problem In Vertex Form:

Example: h=-0.2(d-10)2 + 20

## 2. Identifying Quadratic Relations in factored form: y=a(x-r)(x-s) (GRAPHING)

**Example**

**: y=-2(x-1)(x-5):**

__ r__= your first x-intercept (Remember opposite form so -2 is actually +2)

__ s__= your second x-intercept

__ a__= still your vertical stretch/compression. Still reflecting or directing up depending on sign.

__ Axis of symmetry__: Number between the two x-intercepts in this case +3. (+2 & +4)

__ y= __Plug your axis of symmetry into the regular relation as 'x' and expand to find 'y' which will than be your vertex with axis of symmetry. (x,y)

**Therefore:**

x-intercepts are: +1 and +5

vertical stretch is: -2

Axis of symmetry: +3

Finding y:?

y=-2(3-1)(3-5)

y=-2(2)(-2)

y= 8

**So therefore the vertex is (3,8) **

**NOTE: You always sub and isolate for variables x and y. Isolating for x, y=0 and Isolating for y, x=0**

## -What Is Factoring?

For an example: We use a method known as** spreading the rainbow.**

For **y=(x+4)(x+4)**

**We would spread the rainbow we would multiply the first two numbers/variables in each bracket. ( so 'x' multiply by 'x' which gives us x2. Than Add the two outside numbers/variables in each bracket.( 4x plus 4x which is 8x**) **and than multiply the last numbers in each bracket( so 4x4 which is 16) **

**So what were left with is our factored equation of y=x2 + 8x + 16. **

## -Common Factoring

*Common Factoring is when you have an expression and you have to find a Greatest Common Factor and that is placed on the outside of your new and factored expression. Also the you divide each number in the original equation by your GCF and put it in brackets. Your GCF always stays on the outside*

**Example: 3x+9 **

when you divide the numbers trying to find the **GCF **it ends to be **3**. So **GCF=3**

**So your new factored equation: 3(x+3)**

## -Factoring Perfect Squares

**Example: y=(x+4)(x+4) or y=(x+4)2**

**So after Factoring and spreading the rainbow, the equation will be one simple equation in standard form which will be y= x2 + 8x + 16.**

## -Factoring Difference Of Squares

**Example: y=(s-4)(s+4)**

**So final answer would be y= s2 - 16 **

## -Factoring Simple Trinomial

**For an example: x2 + 4x + 6 **

**which will factor into (x )(x ) **

## -Factoring Complex Trinomial

## -Factoring By Grouping

## Word Problem In Factored Form:

**An example a word problem in this form and its solution would be: **

## 3. Quadratic Relations in Standard form: y= ax2+ bx + c

## -Quadratic Formula

*The quadratic formula is a method of finding the x-coordinate of the vertex when there is a standard form equation given. After you have gotten your x-coordinate you may place that x-coordinate back into your original equation and get y, than you have your vertex.*

*NOTE: THE 3 TERMS IN YOUR STANDARD EQUATION ARE REPRESENTED AS ABC IN ORDER. So ax2 is represented as 'a' bx is represented as 'b' and c is represented as 'c'*

*NOTE: YOU WILL HAVE 2 FINAL ANSWERS WHICH CAN REPRESENT X-INTERCEPTS AND YOU HAVE THOSE 2 FROM DOING SUBTRACTING AND ADDING. *

*The quadratic formula is:*

## -Example of Quadratic Formual

## -Completing The Square

Method of doing Completing the square.

## Word Problem In Standard Form:

**For an example: 'If you could factor the expression x2 + kx + 6, what are the possible values that k could have, explain?' **

These types of standard form word problems would be answered like this:

## 4. Me & Quadratics (Reflection)

*When Quadratics was first introduced to us, I was not interested like I was interested when it came to Trig or our other first few units. At first I found Quadratics challenging and annoying just because the methods and graphing in such forms was very confusing. Through out the unit I began to come in for extra help and for guidance how to do these certain methods correctly. After some practice from homework and quizzes these methods and formulas become apart of my daily routine. In some aspects I still find somethings confusing but reviewing and taking the time to understand and practice results to good marks on my final tests.*

## - On my way to improving in Quadratics.

## 5. Connections

**For an example: You have a standard form equation, which than you can break down and simplify it into turn into a factored form equation. ALSO it works the other way around as well. You can have a factored form equation, spread the rainbow and turn that into a standard form equation. **

**Another cool way these forms relate is how you can turn a standard form equation to a vertex form equation by just completing the square.**

## Quadratics In Real Life

For an example, When you are playing basketball and shooting a hoop, the path of your shot can be represented by a quadratics path.

By: Ravi Seeder